Write the sum $LaTeX: \displaystyle -31-25-19 \ldots +167+173$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-25-(-31)=6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -31+(n-1)(6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -31+(n-1)(6)=173 \implies n = 35$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{35} \left(6 n - 37\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 35(-31+173) }{2}=2485$ .